Calculus Examples

$\frac{1}{2} \sin (2 x)$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} \sin (2 x)$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\sin (2 x)]$ .

$\frac{1}{2} \frac{d}{d x} [\sin (2 x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 2 x$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{1}{2} (\frac{d}{d u} [\sin (u)] \frac{d}{d x} [2 x])$

Step 2.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\frac{1}{2} (\cos (u) \frac{d}{d x} [2 x])$

Step 2.3

Replace all occurrences of $u$ with $2 x$ .

$\frac{1}{2} (\cos (2 x) \frac{d}{d x} [2 x])$

Step 3

Differentiate.

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Step 3.1

Combine $\cos (2 x)$ and $\frac{1}{2}$ .

$\frac{\cos (2 x)}{2} \frac{d}{d x} [2 x]$

Step 3.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$\frac{\cos (2 x)}{2} (2 \frac{d}{d x} [x])$

Step 3.3

Simplify terms.

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Step 3.3.1

Combine $2$ and $\frac{\cos (2 x)}{2}$ .

$\frac{2 \cos (2 x)}{2} \frac{d}{d x} [x]$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

$\frac{2 \cos (2 x)}{2} \frac{d}{d x} [x]$

Step 3.3.2.2

Divide $\cos (2 x)$ by $1$ .

$\cos (2 x) \frac{d}{d x} [x]$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\cos (2 x) \cdot 1$

Step 3.5

Multiply $\cos (2 x)$ by $1$ .

$\cos (2 x)$